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If $\lim\limits_{x\to a^+} f(x)=L$ and if $c$ is a function such that $a < c(x) < x$ for all $x > a$, then $\lim\limits_{x\to a^+} f(c(x))=L$. Note: there has been no discussion about continuity or any discussion about the limits of composition functions at this point. Any help would be appreciated!

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For some basic information about writing math at this site see e.g. here, here, here and here. –  Martin Sleziak Oct 10 '12 at 12:19

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Let $\epsilon>0$. Then there exists a $\delta>0$ such that

$|f(x)-L|<\epsilon$ whenever $0<x-a<\delta$. Notice that, $0<c(x)-a<x-a<\delta$, so that in particular, $0<c(x)-a<\delta$ and hence:

$|f(c(x))-L|<\epsilon$

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I was definitely over-analyzing the problem. Your explanation makes a lot of sense.Thanks –  user42864 Oct 10 '12 at 0:30

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